(a) Show that when Laplace’s equation ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 = 0 is written in cylindrical coordinates, it becomes ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 + ∂ 2 u ∂ z 2 = 0 (b) Show that when Laplace’s equation is written in spherical coordinates, it becomes ∂ 2 u ∂ ρ 2 + 2 ρ ∂ u ∂ ρ + cot ϕ ρ 2 ∂ u ∂ ϕ + 1 ρ 2 sin 2 ϕ ∂ 2 u ∂ θ 2 = 0
(a) Show that when Laplace’s equation ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 = 0 is written in cylindrical coordinates, it becomes ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 + ∂ 2 u ∂ z 2 = 0 (b) Show that when Laplace’s equation is written in spherical coordinates, it becomes ∂ 2 u ∂ ρ 2 + 2 ρ ∂ u ∂ ρ + cot ϕ ρ 2 ∂ u ∂ ϕ + 1 ρ 2 sin 2 ϕ ∂ 2 u ∂ θ 2 = 0
Solution Summary: The author explains the Laplace equation in cylindrical coordinates, which is partial, sqrt,theta, z=zendarray.
(a) Show that when Laplace’s equation
∂
2
u
∂
x
2
+
∂
2
u
∂
y
2
+
∂
2
u
∂
z
2
=
0
is written in cylindrical coordinates, it becomes
∂
2
u
∂
r
2
+
1
r
∂
u
∂
r
+
1
r
2
∂
2
u
∂
θ
2
+
∂
2
u
∂
z
2
=
0
(b) Show that when Laplace’s equation is written in spherical coordinates, it becomes
∂
2
u
∂
ρ
2
+
2
ρ
∂
u
∂
ρ
+
cot
ϕ
ρ
2
∂
u
∂
ϕ
+
1
ρ
2
sin
2
ϕ
∂
2
u
∂
θ
2
=
0
Show that the function u(x, y, z) = e3x+4ysin(5z) satisfies the Laplace equation in R3
Verify that the function u = 1/(x2 + y2 + z2)1/2 is a solution of the the three dimensional Laplace equation uxx + uyy + uzz = 0.
An Eulerian flow field is described in Cartesian coordinates by V = 4i+xzj+5y3tk. (a) Is it compressible? (b) Is it steady? (c) Is the flow one-, two- or three-dimensional? (d) Find the y-component of the acceleration. (e) Find the y-component of the pressure gradient if the fluid is inviscid and gravity can be neglected.
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